We present a framework for analyzing shape uncertainty and variability in point-sampled geometry. Our representation
is mainly targeted towards discrete surface data stemming from 3D acquisition devices, where a finite
number of possibly noisy samples provides only incomplete information about the underlying surface. We capture
this uncertainty by introducing a statistical representation that quantifies for each point in space the likelihood
that a surface fitting the data passes through that point. This likelihood map is constructed by aggregating local
linear extrapolators computed from weighted least squares fits. The quality of fit of these extrapolators is combined
into a corresponding confidence map that measures the quality of local tangent estimates. We present an analysis
of the effect of noise on these maps, show how to efficiently compute them, and extend the basic definition to a
scale-space formulation. Various applications of our framework are discussed, including an adaptive re-sampling
method, an algorithm for reconstructing surfaces in the presence of noise, and a technique for robustly merging a
set of scans into a single point-based representation.